Below is a tentative list of homework problems for Math 150. In general, HW is due at the start of each class, and we will typically cover on the order of one section a day. All problems are worth 10 points; I will drop whatever assignment helps your HW average the most.  Homework solutions are available here.

HOMEWORK: Homework problems listed below; suggested problems collected together at the end. Note dates MAY change (original dates are from 2018)

• Due Monday, Feb 22 : Handout from first day of class is here   Videos: 2/7/20: Sec 1:   https://youtu.be/gfeXKy3fk1M
  • Read: Section 11.1, 11.2. Use that time to read the material and make sure your Calc I/II is fresh. Feel free to check out the review videos: part 1  part 2. I am trying to get a rotated version posted on glow.
  • Video of the week: Fibonacci numbers (click here for more on the Fibonacci numbers).
  • Slides on the course mechanics.
  • HW problems: (1) What is wrong with the following argument (from Mathematical Fallacies, Flaws, and Flimflam - by Edward Barbeau): There is no point on the parabola 16y = x² closest to (0,5). This is because the distance-squared from (0,5) to a point (x,y) on the parabola is x² + (y-5)². As 16y = x² the distance-squared is f(y) = 16y + (y-5)². As df / dy = 2y+6, there is only one critical point, at y = -3; however, there is no x such that (x,-3) is on the parabola. Thus there is no shortest distance! (2) Compute the derivative of cos(sin(3x² + 2x ln x)). Note that if you can do this derivative correctly, your knowledge of derivatives should be fine for the course. (3)  Let f(x) = x² + 8x + 16 and g(x) = x²+2x-8. Compute the limits as x goes to 0, 3 and ∞ of f(x)+g(x), f(x)g(x) and f(x)/g(x).
  • Read Uslan's graduation speech, and email me something about it that resonates with you: http://www.graduationwisdom.com/speeches/0018-uslan.htm (do not include this with the HW you hand in to the TAs).
  • Think about the following: The sum of two numbers is 8 and their product is 15. What is the sum of their reciprocals? How would you solve this?
  • Extra Credit (due 2/12/20):  Assume team A wins p percent of their games, and team B wins q percent of their games. Which formula do you think does a good job of predicting the probability that team A beats team B? Why? (a) (p+pq) / (p+q+2pq),  (b) (p+pq) / (p+q-2pq),  (c) (p-pq) / (p+q+2pq), or (d) (p-pq) / (p+q-2pq). Hand this in to me, NOT to the graders.
• Due Friday, February 26:
  • Read: 11.2, 11.3, 11.4. NOTE: we will skip sections 11.5, 11.6, 11.7
  • HW #2 problems: Section 11.1: Page 823: #9, #18, #38, #42. Section 11.2: Page 833: #1, #39, and also find the cosine of the angle between a = <2, 5, -4> and b = <1, -2, -3>.
• Due Monday, March 1:
  • Read: 11.4, 11.8. NOTE: we will skip sections 11.5, 11.6, 11.7

  • HW #3 problems: Section 11.2: Question 1: The corollary on page 830 states two vectors are perpendicular if and only if their dot product is zero. Find a non-zero vector, say u, that is perpendicular to <1,1,1>.  (Extra credit: find another vector perpendicular to <1,1,1> and the vector u that you just found. This extra credit should be written right after this problem, or as part of this problem.) Question 2: Consider a triangle with sides of length 4, 5 and 6. Which two sides surround the largest angle, and what is the cosine of that angle? Section 11.3: Question 3: Find the determinant of the 2x2 matrix \(\left(\begin{array}{cc}1 & 2 \\
    3 & 4 \end{array}\right)\); in other words, we filled in the entries with the numbers 1, 2, 3 and 4 in that order, row by row. Similarly, find the determinant of the 3x3 matrix \(\left(\begin{array}{ccc}
    1 & 2 & 3 \\
    4 & 5 & 6 \\
    7 & 8 & 9
    \end{array}\right)\); in other words, we fill in the numbers by 1, 2, 3, 4, 5, 6, 7, 8, 9. (Extra credit: find a nice formula for the determinant of the n x n matrix where the entries are \(1, 2, \dots, n^2\) filled as above, and prove your claim. This extra credit should be turned in on a separate sheet of paper.) Question 4: Find the area of the parallelogram with vertices (0,0), (2,4), (1,6), (3,10). FOR FUN -- DO NOT SUBMIT: Here is a great website with 10 excellent commencement speeches. It's worth the time reading these; I particularly liked the one by Uslan (on how it's not enough to just have a good idea, but how to get noticed).

• BONUS LECTURE
  • A mathematica template is available here: MathematicaIntroVer6.nb
  • Mathematica Video Intro (YouTube)        A video of an introduction to Mathematica lecture is available here (file).

• Due Wednesday, March 3:

  • Read: 11.8, 12.1, 12.2.

  • Video of the week:  light cycle scene from Tron (the original).

  • Pictures of the week: lines and art.

  • HW #4 problems: Section 11.3: Page 842: #1, #5, #11, #12. Section 11.4: Page 849: #1, #2, #3, #22.

• Due Friday, March 5:
  • Video: Watch: http://www.youtube.com/watch?v=71_8kHSAE4w   (February 21, 2014: Limits, Defn Partial Derivatives)
  • Read: 12.3, 12.4, 12.5.
  • HW #5: Section 11.8: Page 893: #1, #26. Section 12.2: Page 908: #2, #4, #5, #27, #32. (Extra Credit: Section 11.8: #55.)
• Due Monday, March 8:
  • Video: Watch:  http://www.youtube.com/watch?v=S6wiYRCiQhs   (March 3, 2014: Derivatives)
  • Read 12.4, 12.5, 12.6, 12.7.
  • HW #6: Section 12.3: Page 917: #1, #8, #10, #24, #38, #54. Section 12.4: Page 928: #1, #4, #5.

• Due Wednesday, March 10: Midterm Due, no video REQUIRED to watch

  • There is a video to watch of an introductory lecture I gave (http://www.youtube.com/watch?v=g1oj7CIqGM8), and a template (http://web.williams.edu/Mathematics/sjmiller/public_html/math/LaTexMathematica/MathematicaIntroVer6.nb). There are more links to these on my handouts page http://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm, For a detailed description of Mathematica see the solutions to HW 5, http://web.williams.edu/Mathematics/sjmiller/public_html/150/hwsolns/HWSolns_Math150_Sp2014.pdf .

  • BONUS LECTURE: Programming: Sec 1: https://youtu.be/2xmH6w7xdyo    Sec 2: https://youtu.be/2xmH6w7xdyo

• NEED TO FIX DATES FROM HERE ON DOWN
• Due Monday, Mar 2:
  • Video: Watch: http://www.youtube.com/watch?v=L1WUWmjvnE0   (March 5, 2014: Optimization)
  • Read 12.4, 12.5, 12.6, 12.7. Read my notes on the Method of Least Squares and Sections 12.6, 12.7.
  • Video of the week: Flatland trailer. (The full movie is available here for class purposes only.) There's also projections of 4-dimensional cubes in our 3-dimensional space.
  • HW: Due Wednesday March 4: Section 12.4: Page 928: #21, #25, #33, #36, #63 (is this surprising?).
• Due Wednesday, Mar 4:
  • Video: Watch  http://youtu.be/Da0cO905Aj8  (one of my favorite lectures in the course; we will NOT cover this material in class to free up time to do more optimization problems)
  • Videos of the week: The Battle of Trafalgar: Animated battle clip    Facts on the battle    Best animated re-enactment I could find  (For more on this see the Additional Comments.)
  • HW: Due Monday Mar 9: Section 12.5: Page 940: #5, #11, #29, #61.
  • Doing more optimization problems in class
• Due Friday, March 6:
  • Video: Watch  http://youtu.be/Da0cO905Aj8  (one of my favorite lectures in the course)
  • Videos of the week: The Battle of Trafalgar: Animated battle clip    Facts on the battle    Best animated re-enactment I could find  (For more on this see the Additional Comments.)
  • HW: Due Monday Mar 9: Section 12.5: Page 940: #5, #11, #29, #61.

• Due Monday, March 9:

  • Video: No video to watch
  • Read 12.4, 12.5, 12.6 and my notes on the Method of Least Squares.
  • HW: Due Wednesday March 11: Homework from handout http://www.williams.edu/Mathematics/sjmiller/public_html/105/handouts/MethodLeastSquares.pdf:  Exercise 3.3: Consider the observed data \((0,0), (1,1), (2,2)\). Show that if we use (2.10) from the Least Squares handout to measure error then the line \(y=1\) yields zero error, and clearly this should not be the best fit line! Exercise 3.9: Show that the Method of Least Squares predicts the period of orbits of planets in our system is proportional to the length of the semi-major axis to the 3/2 power.
• Due Wednesday, March 13:
  • Read 12.7, 12.8
  • Video: Watch http://youtu.be/cgIoKYr11sY   (March 12, 2014: Chain Rule)
  • Homework: (1) Page 949: #18: Use the exact value of \(f(P)\) and the differential \(df\) to approximate the value \(f(Q)\), where \(f(x,y)=\sqrt{x^2-y^2}\), with points \(P(13,5)\) and \(Q(13.2,4.9)\). (2) Page 949: #23: Use the exact value of \(f(P)\) and the differential \(df\) to approximate the value \(f(Q)\), where \(f(x,y,z)= e^{-xyz}\) with the points \(P=(1,0,-2)\) and \(Q=(1.02, 0.03, -2.02)\). (3) Briefly describe what Newton's Method is used for, and roughly how it works.
  • Extra Credit: to be handed in on a separate paper: Let \(f(x) = \exp(-1/x^2)\) if \(|x| > 0\) and 0 if \(x = 0\). Prove that \(f^{(n)}(0) = 0\) (i.e., that all the derivatives at the origin are zero). This implies the Taylor series approximation to \(f(x)\) is the function which is identically zero. As \(f(x) = 0\) only for \(x=0\), this means the Taylor series (which converges for all \(x\)) only agrees with the function at \(x=0\), a very unimpressive feat (as it is forced to agree there).
• Due Friday, March 16:
  • Read 12.8, 12.9
  • Video: Watch: Extending the Pythagorean Formula: https://youtu.be/EH6PUS2OwUY (slides here)
  • Homework: Due Monday March 16: Note: the notation for this homework is a bit annoying. For example, imagine we have a function \(f:\mathbb{R}^3\to\mathbb{R}\) and \(x, y, z:\mathbb{R}^2 \to \mathbb{R}\), so we have \(A(u,v) = f(x(u,v), y(u,v), z(u,v))\). If we want to figure out how this compound function changes with \(u\), I prefer to write \(\frac{\partial A}{\partial u}\); however, the book will often overload the notation and write \(\frac{\partial f}{\partial u}\). I think this greatly increases the chance of making an error, and strongly suggest introducing another function name. Page 960: #2: Find \(dw/dt\) both by using the chain rule and by expressing \(w\) explicitly as a function of \(t\) before differentiating, with \(w = \frac{1}{u^2 + v^2}\), \(u = \cos(2t)\), \(v = \sin(2t)\). Page 960: #5: Find \(\partial w/\partial s\) and \(\partial w/\partial t\) with \(w = \ln(x^2 + y^2 + z^2)\), \(x = s - t\), \(y = s + t\), \(z = 2\sqrt{st}\). Page 960: #8: Find \(\partial w/\partial s\) and \(\partial w/\partial t\) with \(w = yz + zx + xy\), \(x = s^2 - t^2\), \(y = s^2 + t^2\), \(z = s^2t^2\). Page 960: #34: A rectangular box has a square base. Find the rate at which its volume and surface area are changing if its base is increasing at 2 cm/min and its height is decreasing at 3cm/min at the instant when each dimension is 1 meter. Page 960: #41: Suppose that \(w = f(u)\) and that \(u = x + y\). Show that \(\partial w/\partial x = \partial w/\partial y\). 
• DATES FROM HERE ON HAVE NOT BEEN UPDATED
•  
• Due Friday, March 9:
  • Read 12.9, 3.1
  • Homework: Due Friday Mar 9: Page 971: Question 3: Find the gradient \(\nabla f\) at \(P\) where \(f(x,y) = \exp(-x^2-y^2)\) and \(P\) is (0,0). Page 971: Question 10: Find the gradient \(\nabla f\) at \(P\) where \(f(x,y,z) = (2x-3y+5z)^5\) and \(P\) is (-5,1,3). Page 971: Question 11: Find the directional derivative of \(f(x,y) = x^2+2xy+3y^2\) at \(P(2,1)\) in the direction \(\overrightarrow{v} = (1,1)\). In other words, compute \((D_{\overrightarrow{u}}f)(P)\) where \(\overrightarrow{u} = \overrightarrow{v}/|\overrightarrow{v}|\). Page 971: Question 19: Find the directional derivative of \(f(x,y,z) = \exp(xyz)\) at \(P(4,0,-3)\) in the direction \(\overrightarrow{v} = (0,1,-1)\) (which is \(\textbf{j}-\textbf{k}\)). In other words, compute \((D_{\overrightarrow{u}}f)(P)\) where \(\overrightarrow{u} = \overrightarrow{v}/|\overrightarrow{v}|\). Page 971: Question 21: Find the maximum directional derivative of \(f(x,y) = 2x^2+3xy+4y^2\) at \(P(1,1)\) and the direction in which it occurs.
• Due Monday, March 12:
  • Reading: Read 13.1, 13.2
  • Homework Due Mon Mar 12: Question 1: Use Newton's Method to find a rational number that estimates the square-root of 5 correctly to at least 4 decimal places.  Question 2: Let \(w(r,s,t) = f(u(r,s,t), v(r,s,t))\) with \(f(u,v) = u^2 + v^2, u(r,s,t) = t \cos(rs)\) and \(v(r,s,t) = t \sin(rs)\). Find the partial derivatives of \(w\) with respect to \(r\), \(s\) and \(t\) both by direct substitution (which is very nice here!) and by the chain rule. Question 3: Write \((1/2, \sqrt{3}/2)\) in polar coordinates. Question 4: Find the tangent plane to \(z = f(x,y)\) with \(f(x,y) = x^2 y + \sqrt{x+y}\) at \((1,3)\), and approximate the function at \((.9,1.2)\). General comments: These problems are all done the same way. Let's say we have functions of three variables, \(x,y,z\). Find the function to maximize \(f\), the constraint function \(g\), and then solve \(\nabla f(x,y,z) = \lambda \nabla g(x,y,z)\) and \(g(x,y,z) = c\). Explicitly, solve \(\frac{\partial f}{\partial x}(x,y,z) = \lambda \frac{\partial g}{\partial x}(x,y,z)\), \(\frac{\partial f}{\partial y}(x,y,z) = \lambda \frac{\partial g}{\partial y}(x,y,z)\), \(\frac{\partial f}{\partial z}(x,y,z) = \lambda \frac{\partial g}{\partial z}(x,y,z)\), and \(g(x,y,z) = c\). For example, if we want to maximize \(xy^2z^3\) subject to \(x+y+z = 4\), then \(f(x,y,z) = xy^2z^3\) and \(g(x,y,z) = x+y+z = 4\). The hardest part is the algebra to solve the system of equations. Remember to be on the lookout for dividing by zero. That is never allowed, and thus you need to deal with those cases separately. Specifically, if the quantity you want to divide by can be zero, you have to consider as a separate case what happens when it is zero, and as another case what happens when it is not zero. Page 981: Question 1: Find the maximum and minimum values, if any, of \(f(x,y)=2x+y\) subject to the constraint \(x^2+y^2=1\). Page 981: Question 14: Find the maximum and minimum values, if any, of \(f(x,y,z)=x^2+y^2+z^2\) subject to the constraint \(x^4+y^4+z^4=3\).
• Due Wednesday, March 14:
  • Read 13.1, 13.2.
  • For additional reading on some of the background and related material, see the following links. If you're interested in a math major, I strongly urge you to read these.
  •  
    • Intermediate and Mean Value Theorems and Taylor Series (we'll get to the Taylor series part later in the semester).
    • Proofs by Induction (as well as other items, including the notes above!).
  • Homework: Due Wed Mar 14: Page 981: Question 19: Find the point on the line \(3x+4y=100\) that is closest to the origin. Use Lagrange multipliers to minimize the SQUARE of the distance. Page 981: Question 35: Find the point or points of the surface \(z=xy+5\) closest to the origin. Page 981: Question 51: Find the point on the parabola \(y= (x-1)^2\) that is closest to the origin. Note: after some algebra you'll get that \(x\) satisfies \(2(x-1)^3+x=0\) (depending on how you do the algebra it may look slightly different). You may use a calculator, computer program, ... to numerically approximate the solution.
• Due Friday, March 16:
  • Exam in class.... Bring in Problem #1
• Spring Break!
• Due Monday, April 2: REVIEW CLASS
• Due Wednesday, April 4:
  • Read 13.2, 13.3, 13.4.
  • Homework: Due Wednesday, April 4: Page 1004: Question 15: Evaluate \(\int^3_0 \int^3_0 (xy+7x+y) dx dy\). Page 1004: Question 24: Evaluate \(\int^1_0 \int^1_0 e^{x+y} dx dy\). Page 1004: Question 25: Evaluate \(\int^\pi _0 \int^\pi _0 (xy+\sin x) dx dy\). Page 1005: Question 37: Use Riemann sums to show, without calculating the value of the integral, that \(0\leq \int^\pi_0 \int^\pi_0 \sin \sqrt{xy}dxdy\leq \pi^2\). Extra credit: Let \(G(x) = \int_{t = 0}^{x^3} g(t) dt\). Find a nice formula for G'(x) in terms of the functions in this problem.
• Due Friday, April 6:
  • Read 13.3, 13.4. We will not cover 13.5 or 13.6 (though we will of course discuss triple integrals).
  • Video of the week: Coin Sorting. (We talked about this when doing the proof of the Fundamental Theorem of Calculus -- you can integrate by adding as you go, or by grouping by value).
  • Play with Mathematica (or go online to http://www.wolframalpha.com/).
  • Handout with correctly worked example from Monday's class on vertically/horizontally simple region
  • Here are some more problems (with solutions) in setting up double integrals.
  • Homework: Due Friday, April 6: Page 1011: Question 4: Evaluate \(\int_0^2 \int_{y/2}^1 (x+y) dxdy\). Page 1012: Question 11: Evaluate \(\int_0^1 \int_0^{x^3} \exp(y/x)dydx\). Additional Problem: Let \(f(x)=x^3-4x^2+ \cos(2x^3)+ \sin(x+1701)\). Find a finite \(B\) such that \(|f'(x)| \leq B\) for all \(x\) in \([2,3]\). Page 1011: #13: Evaluate the iterated integral \(\int_0^3 \int_0^y \sqrt{y^2 + 16}\ dx\ dy. \). Page 1011: #25: Sketch the region of integration for the integral \(\int_{-2}^2 \int_{x^2}^4 x^2y\ dy\ dx. \) Reverse the order of integration and evaluate the integral. Page 1011: #30: Sketch the region of integration for the integral \(\int_{0}^1 \int_{y}^1 \exp(-x^2)\ dx\ dy. \). Reverse the order of integration and evaluate the integral. Additional Problem: Give an example of a region in the plane that is neither horizontally simple nor vertically simple.
• Due Monday, April 9:  
  • Read 13.7: Just know the statements of cylindrical and spherical change of variables
  • Video of the week: Coin Sorting.
  • Play with Mathematica (or go online to http://www.wolframalpha.com/).
  • Here are some more problems (with solutions) in setting up double integrals.
  • Homework: Due Monday, April 9: Page 1018: #13: Find the volume of the solid that lies below the surface \(z=f(x,y)= y+e^x\) and above the region in the \(xy\)-plane bounded by the given curves: \(x=0\), \(x=1\), \(y=0\), \(y=2\). Page 1018: #42: Find the volume of the solid bounded by the two paraboloids \(z=x^2+2y^2\) and \(z=12-2x^2-y^2\). Page 1026; #13: Evaluate the given integral by first converting to polar coordinates: \(\int_0^1 \int_0^{\sqrt{1-y^2}} \frac{1}{1+x^2+y^2} dx dy.\). Find \(\int_{y = 0}^1 \int_{x=-y}^y x^9 y^8 dx dy\). Additional Question 1: Find \(\int_{y = 0}^1 \int_{x=-y}^y x^9 y^8 dx dy\). Page 1026: Question 4: Evaluate \(\int_{-\pi/4}^{\pi/4} \int_0^{2\cos2\theta} r drd\theta\). Additional Question 2: Evaluate \(\int_0^1\int_{-y}^y \sin(xy) \cdot \exp(x^2y^2) dxdy\). Hint: in what way is this similar to an earlier problem on this homework assignment? Additional Question 3: Let \(f(x,y,z) = \cos(xy + z^2)\). Find D\(f(x,y,z)\). Additional Question 4: Find the maximum value of \(f(x,y) = xy\) given that \(g(x,y) = x^2 + 4 y^2 = 1\).
• Due Wednesday, April 11:
  • Read: 13.9. Know the definition of Jacobian determinants. Read the statement of the Change of Variables formula. We will not deal with this theorem in its full generality, but I want you to at least be aware of its statement. We will concentrate on several special cases: polar coordinates, cylindrical coordinates, and spherical coordinates. If you want more information, click here for my handout on the Change of Variable formula.
• Due Friday, April 13:
  • Read: 13.9. Know the definition of Jacobian determinants. Read the statement of the Change of Variables formula. We will not deal with this theorem in its full generality, but I want you to at least be aware of its statement. We will concentrate on several special cases: polar coordinates, cylindrical coordinates, and spherical coordinates. If you want more information, click here for my handout on the Change of Variable formula.
  • Homework: Due Friday, April 13: Page 1056: #37a. Use spherical coordinates to evaluate the integral \(I \ = \ \int \int \int_B \exp(-\rho^3)\ dV\) where \(B\) is the solid ball of radius \(a\) centered at the origin.  Page 1056: #37b. Let \(a \rightarrow \infty\) in the result of part (a) to show that \(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp(-(x^2 + y^2 + z^2)^{3/2})\ dx\ dy\ dz = \frac{4}{3}\pi.\)
• Due Monday, April 16:
  • Twenty-third day lecture: http://youtu.be/aigdKmu-5ow   (April 28, 2014: Geometric and Harmonic Series, Memoryless Processes)

  • Read multivariable calculus (Cain and Herod) and my lecture notes.

  • Homework: Due Monday April 16: THIS ASSIGNMENT IS ENTIRELY EXTRA CREDIT! IT INVOLVES YOU WATCHING THE VIDEO AND DOING THESE PROBLEMS. IT IS  OPTIONAL. Page 1071: Solve for  \(x\) and \(y\) in terms of \(u\) and \(v\), and compute the Jacobian \(\partial(x,y)/\partial(u,v)\) with \(u = x - 2y, v = 3x + y\). Page 1071: #3: Solve for \(x\) and \(y\) in terms of \(u\) and \(v\), and compute the Jacobian \(\partial(x,y)/\partial(u,v)\) with \(u = xy, v = y/x\).

• Due Wednesday, April 18:
  • Twenty-fourth day lecture:  http://youtu.be/6bf9fjwMs2o   (May 2, 2014: Comparison Test, Implications of Limits of Terms)
  • Read multivariable calculus (Cain and Herod) and my lecture notes.
  • Homework: no written HW due (but the first problem on the exam is due at the start of class, covers material from chapter 11 and chapter 12 but not Lagrange multipliers). Keep reading notes, watch videos to build up a strategic reserve of materials watched, prepare for exam. Remember the first problem is due at the start of class.
• Due Friday, April 20:
  • Twenty-fifth lecture: https://youtu.be/fa4X1AcGFOA (May 2, 2014: Comparison Test, Implications of Limits of Terms: II)
  • Read multivariable calculus (Cain and Herod) and my lecture notes.
  • Homework: Due Friday April 20: Problem 1. Give an example of a sequence \(\{a_n\}_{n = 1}^{\infty}\) that diverges. Problem 2. Give an example of a sequence of distinct terms \(a_n\) such that the sequence \(\{a_n\}_{n = 1}^\infty\) converges. Problem 3. Give an example of a sequence of distinct terms \(a_n\) such that \(|a_n| < 2013\) and the sequence \(\{a_n\}_{n = 1}^\infty\) does not converge. Problem 10-4 (Cain-Herod): Find the limit of the sequence \(a_n = 3/n^2\), or explain why it does not converge. Problem 10-5 (Cain-Herod): Find the limit of the sequence \(a_n = \frac{3n^2+2n-7}{n^2}\).
• Due Monday, April 23: Baseball Lecture
• Due Wednesday April 25: Green's Theorem in a Day
• Due Friday April 27: In-Class Exam: Chapters 11, 12, 13
• Due Monday, April 30:
  • Read multivariable calculus (Cain and Herod) and my lecture notes.
  • Homework: Due Monday April 30: Make sure you understand all the problems from the exam.
• Due Wednesday, April 25:
  • Read multivariable calculus (Cain and Herod) and my lecture notes. (You should have already done this).
  • For Taylor series, see my handout here (essentially just pages 2 and 3).
  • Homework: Dues Wednesday May 2: (1) Cain-Herod: Find the limit of the series \(\sum_{n=1}^\infty \frac{1}{3^n}\). (2) Cain-Herod: Find a value of \(n\) that will insure that \(1+1/2+1/3+\cdots+1/n > 10^6\). Prove your value works. (3) Cain-Herod: Question 14: Determine if the series \(\sum_{k=0}^\infty \frac{1}{2e^k+k}\) converges or diverges. (4) Cain-Herod: Question 15: Determine if the series \(\sum_{k=0}^\infty \frac{1}{2k+1}\) converges or diverges. (5) Let \(f(x)=\cos x\), and compute the first eight derivatives of \(f(x)\) at \(x=0\), and determine the \(n\)-th derivative.
• Due Friday, April 27:
  • For Taylor series, see my handout here (essentially just pages 2 and 3).
  • Homework: Dues Friday May 4: (1) Cain-Herod 10-18: Is the series \(\left(\sum_{k=0}^n\frac{10^k}{k!}\right)\) convergent or divergent? (2) Cain-Herod 10-21: Is the following series convergent or divergent? \(\sum_{k=1}^n \frac{3^k}{5^k(k^4+k+1)}\). (3) Let \(a_n = \frac{1}{(n \ln n)}\) (one divided by \(n\) times the natural log of \(n\)). Prove that this series diverges. \emph{Hint: what is the derivative of the natural log of \(x\)? Use \(u\)-substitution.} (4) Let \(a_n = \frac{1}{ (n\ln^2 n)}\) (one divided by n times the square of the natural log of \(n\)). Prove that this series converges. \emph{Hint: use the same method as the previous problem. (5) Give an example of a sequence or series that you have seen in another class, in something you've read, in something you've observed in the world, ....

• Due Monday, April 30:

  • For Wednesday Watch The Following (note we've already done some of this, such as the integral test)

    • Twenty-fifth day lecture: http://youtu.be/u3BbXey5dVY (May 5, 2014: Ratio Test, Patton Movie Clip)

    • Twenty-sixth day lecture: http://youtu.be/ujJbUpCab6M (May 7, 2014: Root Test, Integral Test)

  • Make sure you understand the problems on the exam you got wrong (as always soln keys are available after you hand in the exams). if anything is not clear please talk to the TAs on Sunday or email me. If you talk to classmates make sure they have taken the exam first (some people are sick and taking it late).

  • Homework: Dues Wednesday May 2: (1) Cain-Herod: Find the limit of the series \(\sum_{n=1}^\infty \frac{1}{3^n}\). (2) Cain-Herod: Find a value of \(n\) that will insure that \(1+1/2+1/3+\cdots+1/n > 10^6\). Prove your value works. (3) Cain-Herod: Question 14: Determine if the series \(\sum_{k=0}^\infty \frac{1}{2e^k+k}\) converges or diverges. (4) Cain-Herod: Question 15: Determine if the series \(\sum_{k=0}^\infty \frac{1}{2k+1}\) converges or diverges. (5) Let \(f(x)=\cos x\), and compute the first eight derivatives of \(f(x)\) at \(x=0\), and determine the \(n\)-th derivative.

• STUFF FROM HERE AND BELOW SUBJECT TO CHANGE
• Due Wednesday, May 2: not assigned 2018
  • For Taylor series, see my handout here (essentially just pages 2 and 3).
  • Homework: Due Wednesday, May 2: Watch the streaming video lecture (available here: https://www.youtube.com/watch?v=WjnHNHPhJtU). Write a one or two page summary on it (preferably typed!). You may write about anything you want; natural choices are how multivariable calculus is used in the real world, error correction / detection, dynamic sampling, .... Also write the evaluation for the MSRC.
•  Due Friday, May 4:
  • For Taylor series, see my handout here (essentially just pages 2 and 3).
  • Homework: Dues Friday May 4: (1) Cain-Herod 10-18: Is the series \(\left(\sum_{k=0}^n\frac{10^k}{k!}\right)\) convergent or divergent? (2) Cain-Herod 10-21: Is the following series convergent or divergent? \(\sum_{k=1}^n \frac{3^k}{5^k(k^4+k+1)}\). (3) Let \(a_n = \frac{1}{(n \ln n)}\) (one divided by \(n\) times the natural log of \(n\)). Prove that this series diverges. \emph{Hint: what is the derivative of the natural log of \(x\)? Use \(u\)-substitution.} (4) Let \(a_n = \frac{1}{ (n\ln^2 n)}\) (one divided by n times the square of the natural log of \(n\)). Prove that this series converges. \emph{Hint: use the same method as the previous problem. (5) Give an example of a sequence or series that you have seen in another class, in something you've read, in something you've observed in the world, ....
• Due Monday, May 7:
  • Homework: Due Wednesday, May 9: Cain-Herod: Question 20: Does the series \(\sum_{n=1}^\infty \frac{3^{2k+1}}{10^k}\) converge or diverge? Additional Question 1: Compute the first five terms of the Taylor series expansion of \(\ln (1-x)\) (the natural logarithm of x) about \(x = 0\), and conjecture the answer for the full Taylor series. Additional Question 2: Compute the first five terms of the Taylor series expansion of \(\ln (1+x)\) (the natural logarithm of x) about \(x = 0\), and conjecture the answer for the full Taylor series. Additional Question 3: Give an example of a sequence or series you like. Additional Question 4: Find the second order Taylor series expansion of \(\cos(xy)\) about \((0,0)\). Additional Question 5: Find the second order Taylor Series expansion of \(\cos(\sqrt{x+y})\) about \((0,0)\). Additional Question 6: Find the second order Taylor series expansion of \(\cos(x^3 y^4)\) about \((0,0)\). Extra Credit 1: Give a product of infinitely many distinct, positive terms such that the product converges to a number \(c\) with \(0 < c < \infty\). Extra Credit 2: Let \(\{a_n\}_{n=1}^\infty\) be a sequence of positive numbers such that \(\sum_{n = 1}^\infty 1/a_n\) converges. Let \(B_n = 1/n \sum_{k = 1}^n a_k\). Prove that \(\sum_{n = 1}^\infty 1/B_n\) converges.
• Due Wednesday, May 9:
  • Homework: Due Wednesday, May 9: Cain-Herod: Question 20: Does the series \(\sum_{n=1}^\infty \frac{3^{2k+1}}{10^k}\) converge or diverge? Additional Question 1: Compute the first five terms of the Taylor series expansion of \(\ln (1-x)\) (the natural logarithm of x) about \(x = 0\), and conjecture the answer for the full Taylor series. Additional Question 2: Compute the first five terms of the Taylor series expansion of \(\ln (1+x)\) (the natural logarithm of x) about \(x = 0\), and conjecture the answer for the full Taylor series. Additional Question 3: Give an example of a sequence or series you like. Additional Question 4: Find the second order Taylor series expansion of \(\cos(xy)\) about \((0,0)\). Additional Question 5: Find the second order Taylor Series expansion of \(\cos(\sqrt{x+y})\) about \((0,0)\). Additional Question 6: Find the second order Taylor series expansion of \(\cos(x^3 y^4)\) about \((0,0)\). Extra Credit 1: Give a product of infinitely many distinct, positive terms such that the product converges to a number \(c\) with \(0 < c < \infty\). Extra Credit 2: Let \(\{a_n\}_{n=1}^\infty\) be a sequence of positive numbers such that \(\sum_{n = 1}^\infty 1/a_n\) converges. Let \(B_n = 1/n \sum_{k = 1}^n a_k\). Prove that \(\sum_{n = 1}^\infty 1/B_n\) converges.
• Due Friday, May 11:
  • No written hw: please bring completed blue sheets and SCS forms to class. If you didn't get one, extras are in my mailbox (make sure you grab the form from your section).
  • If you want, you may send me a write-up of the Green's Theorem in a Day lecture (by Monday, May 19th, 11am), and I will grade it out of 140 points and add to your HW average. The video is Lecture 26 on GLOW (http://glow.williams.edu/mod/resource/view.php?id=102886)
  • Possibly Twenty-ninth day lecture:    http://youtu.be/4OcxtpxuSJw (May 14, 2014: Special Series, Alternating Series, Pi formulas, Birthday Problem)

The following are almost surely the assignments, but the dates will change as these are from 2011.   

• Due Friday, May 13, 2011
  • If you plan on coming to the lecture, a good summary is my lecture notes on Green's theorem.
  • The video of the week is a TED lecture (these are great talks, and worth hearing). We'll do Malcolm Gladwell on Spaghetti Sauce (but only a snippet starting around 5 minutes).

Suggested Problems and Extra Credit Problems for Math 105: The suggested problems are not to be turned in, but are for your own personal edification or for additional practice, though of course I and the TAs are happy to chat about these (or any) problems. If you submit an extra credit problem, please clearly mark that it is an extra credit problem.

• Introduction: THREE Extra Credit Problems: (1) Let N be a large integer. How should we divide N into positive integers a_i such that the product of the a_i is as large as possible. Redo the problem when N and the a_i need not be integers. (2) Without using any computer, calculator or computing by brute force, determine which is larger: e^π or π^e. (In other words, find out which is larger without actually determining the values of e^π or π^e). If you're interested in formulas for π, see also my paper A probabilistic proof of Wallis' formula for π, which appeared in the American Mathematical Monthly (there are a lot of good articles in this magazine, many of which are accessible to freshmen). (3) Prove that the product of the slopes of two perpendicular lines in the plane that are not parallel to the coordinate axes is -1. What is the generalization of this to lines in three-dimensional space? What is the analogue of the product of the slopes of the line equaling -1?
• Section 11.1: Page 823: Is #38 true for all points (i.e., if you take any three points in the plane)?
• Section 11.2: Page 833: #59, #61.
• Section 11.3: Page 842: #7, #17a.
• Section 11.4: Page 849: #25, #54, #58, #60.
• Section 11.8: Page 893: #33, #53. Extra Credit: #55.
• Section 12.2: Page 908: #41, #43, #45.
• Section 12.3: Page 917: #41, #51, #55.
• Section 12.4: Page 981: #55, #57, #58, #68.
• Section 12.5: Page 940: #10, #17, #46.
• Section 12.6: Extra Credit: Let f(x) = exp(-1/x²) if |x| > 0 and 0 if x = 0. Prove that f⁽ⁿ⁾(0) = 0 (i.e., that all the derivatives at the origin are zero). Show this implies the Taylor series approximation to f(x) is the function which is identically zero. As f(x) = 0 only for x=0, this means the Taylor series (which converges for all x) only agrees with the function at x=0, a very unimpressive feat (as it is forced to agree there).
• Section 12.7: Page 960: #38, #53.
• Section 12.8: Page 971: #29, #40, #41, #60.
• Section 12.9: Page 981: #36, #37, #47, #49, #62 (important).
• Section 13.1: Page 1004: #33.
• Section 13.2: Page 1011: #41, #44, #49.
• Section 13.3: Page 1018: #29.
• Section 13.4: Page 1026: #7, #34.
• Section 13.7: Page 1056: #47, #48. (Extra credit for solving both of these.)
• Section 13.9: Page 1070: #10, #28, #29.
• From multivariable calculus (Cain and Herod): Exercise 1 (page 10.3). Extra credit: Find a series where the ratio test provides no information on whether or not it converges but the root test says whether or not it converges or diverges.
• Problems leading up to Green's Theorem TBD.